FIG. 1 illustrates cellular network 100 of the prior art. Each cell 102a-c in cellular network 100 is generally defined by an area in which base stations 104a-c are able to communicate with User Equipment (UE) 106a-d. Examples of UEs 106a-d include telephones, smartphones, Personal Data Assistants (PDAs), tablet computers, cellular data devices for use with laptop computers, and the like. Generally, each cell 102a-c includes one corresponding base station 104a-c. 
Any number of UEs 106a-d may be found in cells 102a-c, depending on the use habits of the users of cellular network 100. For example, cell 102b includes two UEs 106c-d. In such an embodiment, both UEs 106c-d may communicate with base station 104b at the same time. Depending on the protocol used by base station 104b, UEs 106c-d may communicate simultaneously, or substantially simultaneously. Alternatively, UEs 106c-d may communicate with base station 104b within a time slot. Additionally, UEs 106a-d may move between cells as the user travels from one area to another. As shown, UE 106a may move from cell 102a to cell 102c. In this sort of case, cell 102a includes two UEs 106a-b initially, but once UE 106a moves to cell 102c, cell 102a only includes UE 106b. Thus cellular networks 100 are generally dynamic in nature, and changes in the topology of cellular network 100 may be random, based upon the user's habits.
FIG. 2 illustrates an example of a topology for any of cells 102a-c. In addition to base station 104, cells 102 may include antenna 202 coupled to base station 104. Antenna 202 receives random access signals from UEs 106 operated by user 204 in a multipath environment and mobile user 206 over one or more Random Access Channels (RACHs) operated by base station 104. The RACH allows UEs 106 to gain initial access to cellular network 100 and facilitates uplink synchronization.
FIG. 3 illustrates RACH detection circuit 300 according to the prior art. RACH detection circuit 300 is often included in base station 104. RACH detection circuit 300 includes CP removal module 302 for removing the Cyclic Prefix (CP) from received symbols. RACH detection circuit 300 also includes downsampling/resampling module 304 for reducing a sample rate to a frequency that is suitable for use by correlator 306. The reduced sample rate simplifies operations of the correlator 306, particularly in FFT module 308. Correlator 306 includes Fast Fourier Transform (FFT) module 308 configured to transform the downsampled symbol and subcarrier demapping module 310, correlator 306 also includes subcarrier demapping module 310 and multiplier 312. Multiplier 312 multiplies the demapped subcarrier with a conjugate of a root sequence in the frequency domain. The multiplied carrier is then converted back to time domain by Inverse Discrete Fourier Transform (IDFT) module 314. In prior systems 300, signature detection and timing offset estimation module 316 detects a random access signal from UE 106 and determines the timing offset of the detected random access signal.
In Long Term Evolution (LTE) mobile communication networks, for example, UEs 106a-d send random access signals to base stations 104a-c, when the UEs 106a-d are in respective cells 102a-c, to gain initial access to cellular network 100. The random access signals may be sampled by the base station at up to 24576 samples (˜1 ms) in the time domain. One problem with the prior art is that it often takes very complex hardware to detect random access signals with high accuracy, particularly with respect to operations performed by FFT module 308.
Although some prior methods and devices do use all 24576 samples to perform random access detection, most systems use some form of down-sampling to reduce hardware complexity. Down-sampling typically involves dividing the number of samples by an integer, and then only selecting a reduced number of signal samples. For example, 2-fold down-sampling would use 12288 samples, and 4-fold down-sampling would only use 6144 samples. Although these down-sampling techniques may reduce hardware complexity, the tradeoff is a performance degradation due to a bias introduced at down-sampling for those specific access signals that do not fall directly on the samples.